##
* *Reflection: Diverse Entry Points
Limiting Your Options - Section 1: Set the Stage

This student's graph clearly shows a lower than expected level of competence. The most noticeable gap is seen by the complete lack of labels or scale. Additionally lower constraints are not marked yet seem to be assumed by the shading.

This sort of work is best discussed directly with students rather than feeding them corrections, so I asked this student about what *x *and *y* represented and what scale he was using. While puzzled at first, as though I was asking a trick question, he eventually responded that x represented the small cups and y the large ones. I suggested that it would help to have those labeled on the axes and also that it's common to include a scale with a graph, or at least to label a few points so the scale is indicated. His response what that he didn't like to add all the "extra stuff" to graphs because it took too long and looked messy. After further discussion I persuaded him to label this and future graphs (at least in my class) by explaining that his graph is just another way of communicating information and if he leaves off labels its like getting a map without cities or distances, - nice to look at but pretty useless.

I chose to share this student conferencing conversation as it reminds me that students at times comply with our directions/requests without understanding the why and the mathematics involved.

# Limiting Your Options

Lesson 6 of 10

## Objective: SWBAT use systems of linear inequalities to find maximum and minimum solutions.

#### Set the Stage

*10 min*

*You will need to have copies of the lemonade information ready for this part of the lesson or be ready to project it on your board. * I begin this lesson by reviewing our discussion yesterday about limits and constraints. I thank my students for their work in helping my friend set up the model she needed to find her most profitable option. I go on to say that we have a new challenge to address today. The question is how much to charge for lemonade to make the best profit. I explain that the lemonade stand sells small and large cups of fresh lemonade. I project the lemonade information on my front board and ask my students to work individually to set up constraints for this problem like they did for others yesterday. **(MP1, MP2, MP4)** While they're working I walk around offering encouragement and reminders as necessary. When everyone is finished I tell them that today we are not just looking for the constraints, we will actually be solving for the optimal result, starting with this problem. I explain that the next step is to graph all the equations and inequalities we've written as constraints, so I distribute graph paper and rulers and let my students get to work. As always, I walk around offering encouragement and redirection as needed. When everyone has completed this section I select one good graph (I look for a graph that is clearly and correctly drawn and labeled) to project and let everyone check their work against it. I tell my students that we're ready to find how many large and how many small cups of lemonade we should sell to make a maximum profit. I circle the points of intersection on the graph and explain that these represent maximum or minimum points within the limits we've set up. You can see the equations and a sketch of the graph on my educreations video.

#### Resources

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#### Put it Into Action

*40 min*

You will need copies of the Linear Programming problems handout for this section of the class. Now that we've worked through an entire linear programming problem as a class, I tell my students that they get to work with their back-partner to solve some problems on their own. I remind them to write the constraints, then graph them, and finally use the points of intersection on the graph to find the desired solution. I distribute the Linear Programming problems handout and ask if there are any questions. **(MP1, MP2, MP4)** While my students are working I walk around offering encouragement and redirection as needed. When everyone is done, I ask them to take a "gallery walk" to check their work against the answers I've posted around the room and ask questions about any they don't understand.

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#### Wrap it Up

*5 min*

I really want my students to remember how important it is to start with good, meaningful equations so I give them a different kind of challenge to wrap up this lesson. I present them with a Linear Programming Challenge all set up with equations, a graph and a solution - except it's the wrong solution. My challenge to my students is that they figure out why it's wrong, where the mistaken assumption is. ** (MP1, MP2, MP6) **This also sets up the lesson for tomorrow, which focuses on whether the solutions are actually viable answers to the problem.

#### Resources

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- UNIT 1: First Week!
- UNIT 2: Algebraic Arithmetic
- UNIT 3: Algebraic Structure
- UNIT 4: Complex Numbers
- UNIT 5: Creating Algebraically
- UNIT 6: Algebraic Reasoning
- UNIT 7: Building Functions
- UNIT 8: Interpreting Functions
- UNIT 9: Intro to Trig
- UNIT 10: Trigonometric Functions
- UNIT 11: Statistics
- UNIT 12: Probability
- UNIT 13: Semester 2 Review
- UNIT 14: Games
- UNIT 15: Semester 1 Review

- LESSON 1: Make It
- LESSON 2: Make it More
- LESSON 3: Double Trouble
- LESSON 4: Going Graphic
- LESSON 5: Out of Bounds
- LESSON 6: Limiting Your Options
- LESSON 7: Does it Work?
- LESSON 8: Twist It
- LESSON 9: Creating Algebraically Review Stations
- LESSON 10: Creating Algebraically; Assessment